Showing papers in "Mathematics of Computation in 1988"

Generation of finite difference formulas on arbitrarily spaced grids

Abstract: On etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d'ordre arbitraire sur des grilles a une dimension d'espacement arbitraire

A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems

Abstract: This paper provides a preconditioned iterative technique for the solution of saddle point problems. These problems typically arise in the numerical approximation of partial differential equations by Lagrange multiplier techniques and/or mixed methods. The saddle point problem is reformulated as a symmetric positive definite system, which is then solved by conjugate gradient iteration. Applications to the equations of elasticity and Stokes are discussed and the results of numerical experiments are given.

Pi and the AGM--A Study of Analytic Number Theory and Computational Complexity.

Abstract: Complete Elliptic Integrals and the Arithmetic-Geometric Mean Iteration. Theta Functions and the Arithmetic-Geometric Mean Iteration. Jacobi's Triple-Product and Some Number Theoretic Applications. Higher Order Transformations. Modular Equations and Algebraic Approximations to pi. The Complexity of Algebraic Functions. Algorithms for the Elementary Functions. General Means and Iterations. Some Additional Applications. Other Approaches to the Elementary Functions. Pi. Bibliography. Symbol List. Index.

Testing a class of methods for solving minimization problems with simple bounds on the variables

Abstract: Description des resultats d'une serie de tests pour une classe de nouvelles methodes du type region de confiance

An Optimal-Order Error Estimate for the Discontinuous Galerkin Method

Abstract: In this paper a new approach is developed for analyzing the discontinuous Galerkin method for hyperbolic equations. For a model problem in R2, the method is shown to converge at a rate 0(hn+l) when applied with nth degree polynomial approxi- mations over a semiuniform triangulation, assuming sufficient regularity in the solution.

An adaptive finite element method for linear elliptic problems

Abstract: On propose une methode adaptative d'elements finis, basee sur une estimation d'erreur optimale de norme maxima pour les problemes elliptiques lineaires

On a new definition of the fractional difference

Abstract: On donne une nouvelle definition de la difference fractionnaire. On etablit plusieurs proprietes (loi exponentielle, loi de Leibniz). On applique les resultats a la solution des equations aux differences lineaires du second ordre

Upper semicontinuity of attractors for approximations of semigroups and partial differential equations

Abstract: : Suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite dimensional system. Conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined. Applications are given to parabolic and hyperbolic partial differential equations.

On the convergence of difference approximations to scalar conservation laws

Abstract: A unified treatment of explicit in time, two level, second order resolution, total variation diminishing, approximations to scalar conservation laws are presented. The schemes are assumed only to have conservation form and incremental form. A modified flux and a viscosity coefficient are introduced and results in terms of the latter are obtained. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first order accurate in general. Convergence for total variation diminishing-second order resolution schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality.

Computing heights on elliptic curves

Abstract: We describe how to compute the canonical height of points orn elliptic curves. Tate has given a rapidly converging series for Archimedean local heights over R. We describe a modified version of Tate's series which also converges over C, and give an efficient procedure for calculating local heights at non-Archimedean places. In this way we can calculate heights over number fields having complex embeddings. We also give explicit estimates for the tail of our series, and present several examples. Let E be an elliptic curve defined over a number field K, say given by a Weierstrass equation (1) y2 +alXy+a3Y = X3 +a2X2 +a4X+a6. The canonical height on E is a quadratic form h: E(K) R. (For the definition and basic properties of h, see [11, VIII, Section 9] or [6, Chapter VI].) The canonical height is an extremely important theoretical tool in the arithmetic theory of elliptic curves, being used for such diverse purposes as studying values of L-functions [5], numbers of integral points [12], and transcendence theory [9]. It is also important as a computational tool, such as its use in Zagier's algorithm for finding integral points up to large bounds [18]. It is thus of interest to have an efficient method for calculating the canonical height of a point. The usual definition of h as a limit h(P) = limn,0 4-nh(x(2nP)) is not practical for computation. Instead, one uses the fact that the canonical height can be written as a sum of local heights, one term for each distinct absolute value on K: (2) h(P) = E nA, (P). vEMK (For example, if K = Q, then MK can be identified with the set of rational primes together with the usual absolute value on Q. The multiplicities nv are chosen so that the product formula holds and so that h is independent of the choice of the field K.) The local height corresponding to a non-Archimedean absolute value is given by intersection theory in a well-known manner. (See, e.g., [2], [4] or [7, Chapter 11, Section 5].) We will describe a quick way to compute non-Archimedean local heights in Section 5. The local height for an Archimedean absolute value is given by a transcendental function, and so efficient computation is somewhat more difficult. J. Tate [15] ?1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page 339 Received August 20, 1987; revised October 21, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 11G05, 14K07, 11D25. *This work was partially supported by NSF grant #DMS-8612393. ** Current address: Mathematics Department, Brown University, Providence, RI 02912. This content downloaded from 207.46.13.103 on Thu, 20 Oct 2016 04:37:48 UTC All use subject to http://about.jstor.org/terms 340 JOSEPH H. SILVERMAN has given an easily computed power series which works for real absolute values. Precisely, for a given curve E and point P = (x, y), he gives a sequence of easily computed numbers co, cl,... so that

A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations

Abstract: A new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested. This method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions. Stability and convergence results are proven for the Chebyshev approximation of linear scalar hyperbolic equations. The eigenvalues of this method applied to parabolic equations are shown to be real and negative.

Graded-mesh difference schemes for singularly perturbed two-point boundary value problems

Abstract: Etude de l'approximation numerique des operateurs differentiels sans points de retournement par des schemas aux differences finies compacts. Analyse de la stabilite

Connection between Gaussian periods and cyclic units

Abstract: This paper finds that all known parametric families of units in real quadratic, cubic, quartic and sextic fields with prime conductor are linear combinations of Gaussian periods and exhibits these combinations. This approach is used to find new units in the real quintic field for prime conductors p n4 + 5n3 + 15n2 + 25n + 25.

The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems

Abstract: We prove some new estimates for the convergence of multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems. We provide results for the so-called 'symmetric' multigrid schemes. We show that for the variable V-script-cycle and the W-script-cycle schemes, multigrid algorithms with any amount of smoothing on the finest grid converge at a rate that is independent of the number of levels or unknowns, provided that the initial grid is sufficiently fine. We show that the V-script-cycle algorithm also converges (under appropriate assumptions on the coarsest grid) but at a rate which may deteriorate as the number of levels increases. This deterioration for the V-script-cycle may occur even in the case of full elliptic regularity. Finally, the results of numerical experiments are given which illustrate the convergence behavior suggested by the theory.

A user-friendly extrapolation method for oscillatory infinite integrals

Abstract: Developpement d'une methode d'extrapolation flexible pour les integrales infinies oscillatoires

A note on piecewise linear and multilinear table interpolation in many dimensions

Abstract: Etude de l'interpolation multilineaire et lineaire par morceaux a plusieurs dimensions a partir de tables. Comparaison des proprietes d'approximation des interpolants

A fast algorithm for the multiplication of generalized Hilbert matrices with vectors

Abstract: On presente un algorithme rapide de complexite logarithmique pour la multiplication des matrices de Hilbert generalisees par des vecteurs

A fast algorithm for linear complex Chebyshev approximations

Abstract: We propose a new algorithm for finding best minimax polynomial approximations in the complex plane. The algorithm is the first satisfactory generalization of the well-known Remez algorithm for real approximations. Among all available algorithms, ours is the only quadratically convergent one. Numerical examples are presented to illustrate rapid convergence.

Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers

Abstract: It is shown that the binary expansions of algebraic numbers do not form secure pseudorandom sequences, given sufficiently many initial bits of an algebraic number, its minimal polynomial can be reconstructed, and therefore the further bits of the algebraic number can be computed. This also enables the authors to devise a simple algorithm to factorise polynomials with rational coefficients. All algorithms work in polynomial time

Incompressible finite element methods for Navier-Stokes equations with nonstandard boundary conditions in ³

Abstract: This paper is devoted to the steady state, incompressible Navier-Stokes equations with nonstandard boundary conditions of the form u • n = 0, curl u x n = 0, either on the entire boundary or mixed with the standard boundary condition u = 0 on part of the boundary. The problem is expressed in terms of vector potential, vorticity and pressure. The vorticity and vector potential are approximated with curl-conforming finite elements and the pressure with standard continuous finite elements. The error estimates yield nearly optimal results for the purely nonstandard problem.

Quintic polynomials and real cyclotomic fields with large class numbers

Abstract: We study a family of quintic polynomials discoverd by Emma Lehmer. We show that the roots are fundamental units for the corresponding quintic fields. These fields have large class numbers and several examples are calculated. As a consequence, we show that for the prime p = 641491 the class number of the maximal real subfield of the pth cyclotomic field is divisible by the prime 1566401. In an appendix we give a characterization of the "simplest" quadratic, cubic and quartic fields.

Accurate Boundary ConditiOns for Exterior Problems in Gas Dynamics

Abstract: The numerical solution of exterior problems is typically accomplished by introducing an artificial, far field boundary and solving the equations on a truncated domain. For hyperbolic systems, boundary conditions at this boundary are often derived by imposing a principle of no reflection. However, waves with spherical symmetry in gas dynamics satisfy equations where incoming and outgoing Riemann variables are coupled. This suggests that natural reflections may be important. A reflecting boundary condition is proposed based on an asymptotic solution of the far field equations. Nonlinear energy estimates are obtained for the truncated problem and numerical experiments presented to validate the theory.

Incompressible Finite Element Methods for Navier-Stokes Equations with Nonstandard Boundary Conditions in R 3

Abstract: Application des methodes a elements finis incompressibles aux equations de Navier-Stokes avec des conditions aux limites non standards dans R 3 . Utilisation du potentiel vecteur, de la vorticite et de la pression. Estimations d'erreurs

Product integration-collocation methods for noncompact integral operator equations

Abstract: We discuss the numerical solution of a class of second-kind integral equations in which the integral operator is not compact Such equations arise, for example, when boundary integral methods are applied to potential problems in a two-dimensional domain with corners in the boundary We are able to prove the optimal orders of convergence for the usual collocation and product integration methods on graded meshes, provided some simple modifications are made to the underlying basis functions These are sufficient to ensure stability, but do not damage the rate of convergence Numerical experiments show that such modifications are necessary in certain circumstances